COMPARATIVE STUDY ON LATERAL TORSIONAL BUCKLING OF TUBULAR PLATE GIRDERS USING DIFFERENT WEB SYSTEMS

Abstract

Compared with conventional plate girder with flat web, plate girders with tubular rectangular flanges have shown better flexural resistance and very stiff torsional resistance. Their flat webs are comparatively flexible and may allow for web distortion to reduce their resistances to lateral torsional buckling. Using innovative web systems in plate girders, like corrugated webs, may lead to better distortional resistance and improvement of lateral torsional buckling capacity of plate girders. This paper deals with this problem by comparing different systems of plate girders, different flange and web types, both using finite element model and mathematical model. A simple but sufficiently accurate closed-form solution is introduced for the effect of distortion of simply supported tubular flanges as well as warping resistance for corrugated web. Different types are compared to figure out the best shape and geometrical configuration for the girders.

Full text

E05ST45 - 1
11th ICSGE
17-19 May 2005
Cairo - Egypt
Ain Shams University
Faculty of Engineering
Department of Structural Engineering
Eleventh International Colloquium on Structural and Geotechnical Engineering
COMPARATIVE STUDY ON LATERAL TORSIONAL BUCKLING OF
TUBULAR PLATE GIRDERS USING DIFFERENT WEB SYSTEMS
SHERIF A. IBRAHIM1, PH.D., P.E.
ABSTRACT
Compared with conventional plate girder with flat web, plate girders with tubular rectangular
flanges have shown better flexural resistance and very stiff torsional resistance. Their flat
webs are comparatively flexible and may allow for web distortion to reduce their resistances
to lateral torsional buckling. Using innovative web systems in plate girders, like corrugated
webs, may lead to better distortional resistance and improvement of lateral torsional buckling
capacity of plate girders. This paper deals with this problem by comparing different systems
of plate girders, different flange and web types, both using finite element model and
mathematical model. A simple but sufficiently accurate closed-form solution is introduced for
the effect of distortion of simply supported tubular flanges as well as warping resistance for
corrugated web. Different types are compared to figure out the best shape and geometrical
configuration for the girders.
Keywords: Tubular flange plate girders, Corrugated webs, Lateral torsional buckling, Flange
distortion, Warping constant, hollow flange beams.
1 INTRODUCTION
In recent years, one of the most economic cross sections are these with tubular flanges. They
are more often efficient than conventional plate girders with flat webs. The structural
efficiencies of the tubular flanged plate girders are due to their higher flexural strength as
well as the torsionally rigid closed rectangular flanges.
One of the main objectives of this paper is to investigate the ultimate capacity and strength of
lateral torsional buckling of different cross sections using flat flange plate, flat web plates,
corrugated web plates as well as rectangular tubular flanges. By using these combinations of
flanges and webs, one can reach an optimum profile of the cross section used in laterally
unsupported girders to have maximum resistance against lateral toriosnal buckling.
Using of trapezoidally corrugated webs in plate girders increases the value of warping
constant of such girders (Lindner-1990). He proposed a closed form solution to calculate the
warping constant “Cw” and verify it with experimental tests of girders with corrugated webs
subjected to pure torsion.
___________________________________________________________________________
1Assistant Professor, Ain Shams University, Structural Engineering Dept., Abbasia, Cairo,
Egypt, sai23@drexel.edu

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Unlike the commonly observed lateral torsional buckling of steel girders, the lateral
distortional buckling of tubular flange plate girders is characterized by simultaneous lateral
deflection, twist and cross section change due to web distortion.
The cross section distortion causes significant strength reduction and is particularly severe is
short to medium spans (Mahendran-1997). Lateral distortional buckling affects the overall
lateral torsional buckling strength and is not encompassed by the design formulas contained
in most of the design specifications.
Because of this, the effective torsional buckling rigidity “GJ” maybe reduced substantially
below the normal value. In the case of simply supported beams under uniform bending, this
may cause a significant reduction from the elastic flexural-torsional buckling moment
without web distortion. Therefore an elastic buckling analysis is required to determine its
capacity. Consequently, an investigation was conducted (Trahair-1997) to study the lateral
distortional buckling behavior of tubular flange plate girders to quantify the associated
reduction in flexural strength due to reduction in “GJ”.
One of the ways to eliminate this problem is to use the corrugated web accompanied with the
tubular flange since it give higher tosrional stiffness than the flat web as shown in figure (1).
2 GIRDERS DESIGN AND INVESTIGATION
Based on comprehensive study by the author, four profiles of plate girders were shown to be
investigated, mainly, plate girders with flat webs and flanges, plate girders with corrugated
webs and flat flanges, plate girders with flat webs and tubular flanges and plate girders with
corrugated webs and tubular flanges as shown in figure (2).
The dimensioning of different models are shown in table (1) where three families are set, one
with slender flat web 60x0.4 cm and using compact flange plate or tubular flanges with the
same flange area as shown in table. The second with the same flange arrangement as the first
but with corrugated web. The profile of the trapezoidally corrugated webs used in the study is
shown in figure (3). The length of the horizontal fold “b” is chosen 20cm and the length of
the inclined fold projection “a” is chosen 5 cm, whereas the thickness of the corrugated web
is kept 0.4 cm allover the study. These parameters were kept constant in the research based
on parametric study by the author where they were found to be ineffective on the lateral
torsional buckling of plate girders with corrugated webs.
The main effective parameter in the corrugation profile parameters is found to be the depth of
corrugation “hr”. In this paper the depth of corrugation is 5 cm in the second family, or hr/bf
=0.5, and in the third family with wide corrugation depth hr= 10 cm or hr/bf =1, so that we
can study the effect of corrugation depth with the tubular flange as well.
The girders modeled were chosen as simply supported with total length L=5.0 m and loaded
under four points loading condition in quarter and three quarters of the length so that we have
half of the span under uniform flexural moment. The moment gradient factor “Cb” in this case
is 1.0 so it simulates the case of beam under uniform bending moment.
3 FINITE ELEMENT ANALYSIS
3.1 Finite Element Model
For this study, the finite element analysis program (COSMOS/M -2.6) was used to model the
pre-mentioned plate girders with different web and flanges systems. As indicated before, each
girder is modeled as a simply supported beam under four points loading which is simulated in
the program as a line load across the flange width. Lateral supports are only provided at the
ends of the upper flanges whereas the length of the girder is kept laterally unsupported.

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Transverse stiffeners are provided at the girder ends, over the supports, with a thickness of 4
cm to avoid any encountering buckling problem in the stiffeners itself. Additional stiffeners
are provided as well inside the tubular flanges under the vertical loads location to prevent
local buckling of the tubular flanges.
The top and bottom flanges are modeled using four elements per flange width. Twenty four
elements are used to model the height of the web in all cases. The horizontal fold is modeled
using eight elements while the inclined fold is modeled using two elements. The aspect ratio
of any shell element did not exceed 2 so as to obtain accurate results.
The finite element mesh is shown in figure (5) and the used element type from the program
library is the four nodes thin shell element SHELL4T. A nonlinear static analysis is
performed considering the effects of geometrical and material non-linearities and using the
automatic Arc-Length control technique of the program. Modified Newton-Raphson solution
technique is adapted in these models as well.
3.2 Material Properties
The steel used in both the flanges and the webs is assumed high grade steel 36/52 with a yield
stress (Fy) 3.6 t/cm2. For the stiffeners, the steel properties used are assumed to be the same as
the flange material. The typical stress-strain relationship used in the models is assumed to be
bi-linear relation as shown in figure (4). The young’s modulus of elasticity (E) is assumed to
be 2100 t/cm2 up to the yield stress and 21 t/cm2 (E/100) in the post yield range. This stress-
strain relationship is used instead of the linearly-elastic perfectly-plastic relation to avoid
encountering problems with convergence of the solution.
4 RESULTS OF FINITE ELEMENT ANALYSIS
Based on the obtained results, most of the plate girders did not exhibit post buckling strength
for the lateral torsional or lateral distortional buckling mode of failure, except specimens
CWG-H6 and CWG-W-H6, therefore, elastic buckling was deemed to be the dominant and
most appropriate method of analysis.
Figure (5) shows a typical failure mode by lateral torsional buckling accompanied by
distortion of the compression flange. This failure shape was common for all the models in our
study except for models with flat flange plates where only lateral torsional buckling was the
dominant failure mode.
Figure (6) shows the moment versus lateral-displacement of compression flange for flat web
systems with different flanges. The obtained ultimate moment capacity from the finite
element is shown in Table (2). All the specimens in the group failed due to lateral torsional
buckling in combination with lateral distortional buckling mode. It is obvious that using the
tubular flanges increases the flexural moment capacity of the girder up to 40%, as shown in
Table (2), although all the systems have the same cross-sectional area.
Whereas, figures (7) and (8), show the same relation for the same flanges systems
accompanied with the corrugated web system with depth of corrugation 5 cm and 10 cm,
respectively. Using the corrugated web in the plate girder leads to increase to the flexural
moment capacity by about 87%. This means another increase of 47% due to the use of
corrugation with the tubular flanges. This jump in the capacity is due to the high stiff
corrugated web in torsion which reduces the distortion of the tubular flange out-of-plane and
consequently increasing the moment capacity of the cross section. Using of wide corrugation
depth results in an increase by 11% when comparing model CWG-W to CWG and by 32%
when comparing model CWG-W-H2 to CWG-H2. For specimens CWG-H6 and CWG-W-
H6, yielding starts at some minor zones in-between the loading points.

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Figure (9) shows the moment versus lateral-displacement of compression flange for flat
flange plate with different web systems. Using plate girders with corrugated webs leads to an
increase of 20% due to the increase in out-of-plane stiffness. In figure (10), the percentage
increased to 73% with the use of tubular flanges (10x2x0.4) because of the increase of the
torsional stiffness of the cross section. Figure (11) shows the moment versus lateral-
displacement of compression flange for tubular flange (10x6x0.3) with different web systems
and the failure was accompanied by yielding of the flanges (My=21.04 m.t).
5 MATHEMATICAL MODEL
5.1 Effective Torsional Rigidity
Since the webs are torsionally weak compare to their flanges and may allow for web
distortion to reduce their resistance to flexural-torsional buckling, the effective torsional
buckling GJ maybe reduced below the nominal value to GJr and this may cause a reduction
for elastic lateral torsional buckling moment resistance without web distortion.
The flanges of I–section are not stiff torsionally, so web distortion does not become very
significant unless the web is particularly slender or only the tension flange is restrained
torsionally. Trahair proposed an approximation for the reduced effective torsional buckling
GJr as follows
GJr=
hπ91.0
LEt
)2/π(+GJ2
hπ91.0
LEt
)2/π(
2
23
2
2
23
2
(1)
Where E Young’s modulus of elasticity =2100 t/cm2
t thickness of web plate (cm)
h depth of web plate (cm)
L Span length of girder (cm)
This value of reduced torsional buckling is used in the elastic lateral torsional buckling
moment resistance equation as will be shown later.
5.2 Warping Constant Modifications
Lindner (1990), proposed a formula for the warping constant of plate girders with
corrugated webs (Cw*). Then he described a solution for the problem of lateral torsional
buckling of such girders. The modified warping constant (Cw*) is calculated from
Cw* = Cw + Iw (L2/ 2 E) (2)
Iw =
)a+b(β8
h.h 22
r
(3)
3
ff
2
32
tEbb25
)a+b(h
+
Gbt2
h
=β
(4)
Where hr depth of corrugation (cm)
b horizontal fold length (cm)
a horizontal projection of the inclined fold (cm)
bf, tf flange width and thickness, respectively (cm)

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Cw warping constant (cm6) for flat web = Iy h2 / 2
This warping constant consists of two parts:
1) The first part is the same value of warping constant for flat web, which is dependent on
the moment of inertia about the strong axis of the compression flange without any
contribution from the web.
2) The second part is the contribution of the corrugated web which is dependent on the
corrugation depth “hr”.
5.3 Modified Lateral Torsional Buckling Capacity
Based on the previous approach for calculating the elastic lateral torsional buckling moment
resistance, the reduced effective torsional buckling GJr can be used when the plate girder
cross-section has a tubular flange. As well, the modified warping constant Cw* can be used
when the plate girder has a corrugated web. Consequently, the mathematical model for the
elastic lateral torsional buckling moment resistance of simply supported plate girder with
tubular flanges and corrugated web under uniform bending moment can be calculated from
the equation
Mcr =
*
wy
2
b
ry
b
CI
L
Eπ
+GJEI
L
π

Mp (5)
Where Mp the plastic moment capacity of the section. (t.cm)
6 COMPARISON BETWEEN FINITE ELEMENT MODEL AND MATHEMATICAL
MODEL
This part includes a comparison between the finite element results and the values obtained
from the mathematical model used, from equation (1) to (5), to validate the accuracy of
mathematical model to solve the problem of lateral torsional and lateral distortional buckling
of girders having any of the cross sections mentioned before. Table (2) includes this
comparison and it shows good correlation between the finite element and the mathematical
model for elastic lateral torsional buckling. The average ratio of MF.E/ MM.M is 1.062 in favor
of finite element results. It is worth mentioning that for girders CWG-H6 and CWG-W-H6,
and based on equation (5), these girders reached the yield moment capacity of 21.04 m.t and
it was comparable with the finite element result, namely, 0.965 and 0.957 for the two
specimens as shown in table (2), respectively. This can be considered a good agreement
between the two methods.
7 CONCLUSION
Elastic lateral torsional and distortional buckling have been investigated for different types of
plate girders with different web systems. These systems such as rectangular tubular flanges or
trapezoidally corrugated webs show a considerable contribution to the elastic flexural-
torsional moment capacity of simply supported plate girders. Using rectangular tubular flange
accompanied with corrugated web can achieve more than 80% more than the corresponding
conventional plate girder with the same cross sectional area and consequently the same
weight. A mathematical model is proposed and verified with the finite element analysis and
taking into consideration the effect change of torsional rigidity as well as warping constant

E05ST45 - 6
change in the mathematical model. They showed good agreement and average results of
1.062.
8 REFERENCES
[1] Lindner, J. (1990), “Lateral Torsional Buckling of Beams with Trapezoidally Corrugated
Webs”, Proceedings of 4th International Colloquium Stability of Steel Structures, Hungary.
[2] Mahendran, M. and Avery, P. (1997) “Finite-Element Analysis of Hollow Flange Beams
with Web Stiffeners”, Journal of the Structural Division (ACSE), v. 123, No.9, Sept.97.
[3] Mahendran, M. and Avery, P. (1997) “Buckling Experiments on Hollow Flange Beams
with Web Stiffeners”, Journal of the Structural Division (ACSE), v. 123, No.9, Sept.97.
[4] Trahair, N. and Pi, Y. (1997) “Lateral-Distortional Buckling of Hollow Flange Beams”,
Journal of the Structural Division (ACSE), v. 123, No.6, June 97.
[5] Lindner, J. and Haung, B. (1994), “Progress in the Analysis of Beams with Trapezoidally
Corrugated Webs”, Proceedings of 17th Czech and Slovak Conference on Steel Structures and
Bridges, Hungary.
[6] Lindner, J. (1991), “Load Carrying Capacity of Beams Subjected to Local Plate Buckling
and Overall Lateral Torsional Buckling”, Proceedings of 4th International Colloquium on
Structural Stability- Mediterranean Session, Istanbul.
[7] Lindner, J. and Aschinger, R. (1990), “Torsional Stiffness of Welded I-Girder with
Trapezoidally Corrugated Webs”, Proceedings of 4th International Colloquium Stability of
Steel Structures, Hungary.
[8] Salmon, C. G. and Johnson, J. E., “Steel Structures: Design and Behavior”, 4th edition,
Harper Collins, New Jersey, 1998.

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Table 1: Designation of plate girders with different types of flange and web system
Symbol
Designation
Flat
Flat web (60x0.4) cm – Flanges plates (10x1.0) cm
Flat-H2
Flat web (60x0.4) cm – Tubular Flange (10x2x0.4) cm
Flat-H6
Flat web (60x0.4) cm – Tubular Flange (10x6x0.3) cm
CWG
Corrugated web (t=0.4 – hr=5 cm) – Flanges plates (10x1.0) cm
CWG-H2
Corrugated web (t=0.4 – hr=5 cm) – Tubular Flange (10x2x0.4) cm
CWG-H6
Corrugated web (t=0.4 – hr=5 cm) – Tubular Flange (10x6x0.3) cm
CWG-W
Corrugated web (t=0.4 – hr=10 cm) – Flanges plates (10x1.0) cm
CWG-W-H2
Corrugated web (t=0.4 – hr=10 cm) – Tubular Flange (10x2x0.4) cm
CWG-W-H6
Corrugated web (t=0.4 – hr=10 cm) – Tubular Flange (10x6x0.3) cm
Table 2: Comparison of moment capacity of plate girders between finite element models and
mathematical models.
Model
MF.E
MM.M
MF.E/ MM.M
MF.E / MFlat (F.E)
Flat
10.825
10.05
1.077
1.0
Flat-H2
12.75
11.02
1.157
1.178
Flat-H6
15.1875
13.268
1.145
1.403
CWG
11.741
10.36
1.133
1.085
CWG-H2
14.175
14.45
0.981
1.31
CWG-H6
20.296
21.04**
0.965
1.875
CWG-W
13.04
12.04
1.083
1.205
CWG-W-H2
18.725
17.67
1.06
1.73
CWG-W-H6
20.125
21.04**
0.957
1.86
Avr. = 1.062
** The moment capacity due to yielding of the flanges.

E05ST45 - 8
cwg-h
Flat Flat-h
cwg
Fig.(1): Tubular flange plate girder Fig.(2): Types of different investigated cross sections.
with corrugated web.
b
ab Or
ht
Fy=3.6 t/cm2
E=2100 t/cm2
Et=21 t/cm2 (E/100)
Strain (%)
Et
Fy
0
E
Stress
Fig.(3): Corrugated web profile Fig.(4): Typical stress-strain relationship
used in the finite element models
Fig.(5): Failure mode by lateral torsional buckling of finite element model CWG-H2

E05ST45 - 9
0
2
4
6
8
10
12
14
16
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6
Lateral deformation (cm)
Ultimate moment (m-ton)
Flat
Flat-H2
Flat-H6
Fig.(6): Moment versus lateral-displacement of compression flange for flat web systems
0
3
6
9
12
15
18
21
0 1 2 3 4 5 6 7 8 9 10 11 12 13
Lateral deformation (cm )
Ultimate moment (m-ton)
CWG
CWG-H2
CWG-H6
Fig.(7): Moment versus lateral-displacement of compression flange for corrugated web
systems - hr =5 cm
0
3
6
9
12
15
18
21
0 1 2 3 4 5 6 7 8 9
Lateral deformation (cm)
Ultimate moment (m-ton)
CWG-W
CWG-W-H2
CWG-W-H6
Fig.(8): Moment versus lateral-displacement of compression flange for corrugated web
systems – Wide corrugation depth - hr =10 cm

E05ST45 - 10
0
2
4
6
8
10
12
14
0 1 2 3 4 5 6 7 8 9
Lateral deformation (cm)
Ultimate moment (m-ton)
Flat
CWG
CWG-W
Fig.(9): Moment versus lateral-displacement of compression flange for various web systems
– Flanges plates (10x1.0) cm
0
2
4
6
8
10
12
14
16
18
20
0 1 2 3 4 5 6 7 8 9 10
Lateral deformation (cm)
Ultimate moment (m-ton)
Flat-H2
CWG-H2
CWG-W-H2
Fig.(10): Moment versus lateral-displacement of compression flange for various web systems
– Tubular Flange (10x2x0.4) cm
0
3
6
9
12
15
18
21
0 1 2 3 4 5 6 7 8 9 10 11 12 13
Lateral deformation (cm)
Ultimate moment (m-ton)
Flat-H6
CWG-H6
CWG-W-H6
Fig.(11): Moment versus lateral-displacement of compression flange for various web systems
– Tubular Flange (10x2x0.4) cm